UNIVERSITY  OF  CALIFORNIA   PUBLICATIONS 

IN 

AGRICULTURAL    SCIENCES 

Vol.  3,  No.  4,  pp.  55-61  November  27,  1917 


A  NEW  DENDROMETER 

BY 

DONALD  BRUCE 


There  is  a  growing  demand  for  a  satisfactory  dendrometer,  or 
instrument  which  will  measure  the  diameter  of  trees  at  points  out  of 
reach  from  the  ground.  An  indication  both  of  the  wide  demand  and 
of  the  requirements  which  such  an  instrument  must  meet  may  be 
gained  from  a  consideration  of  the  following  instances. 

In  certain  regions,  United  States  Forest  Service  timber  estimators 
have  made  use  of  a  volume  table  based  on  a  diameter  measurement 
at  the  top  of  the  first  sixteen-foot  log  instead  of  at  the  conventional 
breast-high  point.  This  was  on  account  of  the  abnormal  form  of  the 
badly  burned  butts,  which  made  a  lower  measurement  both  uncertain 
and  a  poor  index  of  volume.  Considerable  trouble  resulted  through 
inability  to  check  the  ocular  estimates  of  diameters  except  by  uncer- 
tain methods  of  measuring  at  breast  height  and  subtracting  the  esti- 
mated taper.  For  such  cases  there  is  needed  a  dendrometer  of 
moderate  precision,  large  range,  considerable  rapidity,  lightness,  and 
portability. 

Many  volume  tables  are  based  on  a  measurement  of  height  to  a 
certain  fixed  cutting  limit,  such  as  six,  eight,  or  ten  inches  top 
diameter.  From  the  ground  it  is  often  more  difficult  to  identify  this 
point  than  it  is  to  estimate  its  height,  and  considerable  errors  result. 
Instruments  of  only  a  small  range  of  sizes  are  needed,  and  in  fact  for 
a  given  volume  table,  or  a  consistent  set  of  tables,  an  instrument  that 
can  be  fixed  and  adjusted  for  a  single  diameter  would  serve  the 
purpose. 

Other  volume  tables  are  based  on  height  to  the  limit  of  merchant- 
ableness.  This  limit,  however,  varies  widely  in  different  regions,  even 
for  a  single  species,  and  to  use  such  a  volume  table  accurately  one 
must  know  the  top  diameter  corresponding  with  each  value  of  the 
table  and  estimate  heights  accordingly.     Exactly  the  same  type  of  in- 


56 


University  of  California  Publications  in  Agricultural  Sciences       [Vol.  3 


strument  is  required  as  in  the  last  case,  save  that  a  slightly  larger 
range  of  diameters  is  needed  and  a  fixed  adjustment  for  a  definite  size 
is  not  adequate. 

Many  Pacific  coast  volume  tables  are  based  on  diameter,  height, 
and  taper.  While  the  first  two  factors  are  measured,  at  least  occasion- 
ally, the  last  is  usually  a  matter  of  guesswork  entirely.  The  instru- 
ment needed  to  strengthen  this  part  of  the  work  is  a  dendrometer 
possessing  the  qualities  above  mentioned,  and  in  addition  one  which 
works  independently  of  distance,  since  both  horizontal  distances  and 
heights  will  usually  be  but  roughly  approximated. 

In  many  scientific  studies  of  growth  on  permanent  sample  plots  in 
this    country   periodic    measurements    of    diameters    breast-high    and 


&**<t/t/rnr  , 


J&hdi'ay/'f'mi' 


2>/recfL,ne    o£_SJsAL 


Jnctisect Z//7e  of-  S/jhf 


Fig.  1 

heights  are  being  secured,  and  growth  in  volume  is  being  calculated 
from  these  data  by  means  of  a  single  volume  table  for  each  species. 
As  a  result,  whatever  growth  results  from  a  change  in  tree  form  is 
being  neglected.  A  dendrometer  is  needed  of  considerable  precision, 
but  not  necessarily  so  portable  or  rapid  in  action  as  in  the  previous 
cases.  Its  range  in  most  cases  need  not  be  great,  since  the  more 
important  growth  problems  are  connected  with  second-growth  timber 
or,  at  least,  with  trees  below  a  certain  diameter  limit. 

Schiffel's  formula  for  obtaining  volume  has  not  been  sufficiently 
tested  for  most  American  species,  but  it  is  regarded  as  probably  having 
a  high  value  in  many  cases.  It  requires  a  measurement  of  diameter 
at  a  point  half  way  up  the  bole,  and  hence  a  dendrometer.  The 
qualifications  of  a  satisfactory  instrument  will  naturally  depend  on 
the  character  of  the  work  being  done. 

All  these  instances  indicate  that  it  is  not  due  to  the  absence  of  a 
real  need  that  dendrometers  are  practically  unknown  in  America.  It 
seems  obvious,   rather,  that  no  existing  type  satisfies  the  conditions 


1917]  Bruce:  A   New  Dendrometer  57 

above  outlined.  The  following  pages  describe  an  instrument  based  on 
a  somewhat  different  principle  from  those  previously  devised,  which 
will  be  seen  in  a  large  measure  to  meet  these  requirements. 

It  consists  essentially  of  a  straight  arm  upon  which  are  mounted 
two  small  mirrors,  both  at  an  angle  of  45  degrees  with  the  axis  of  the 
arm,  parallel  to  each  other  and  facing  in  opposite  directions  (see 
fig.  1).  One  mirror  is  fixed  at  one  end  of  the  arm,  while  the  other 
is  mounted  on  a  slide  which  travels  along  the  arm.  Graduations 
permit  a  direct  reading  of  the  distance  between  the  mirrors. 

The  principle  is  indicated  by  figure  1  which  shows  the  relative 
position,  as  seen  from  above,  of  tree,  observer's  eye,  and  of  the  instru- 
ment when  in  use.  It  will  readily  be  seen  that  the  instrument  is 
closely  akin  to  the  ordinary  calipers  in  principle,  except  that  for  the 
parallel  fixed  and  movable  arms  of  the  calipers  are  substituted  two 
parallel  lines  of  sight.  The  direct  line  of  sight  passes  just  above  the 
upper  edge  of  the  fixed  mirror  from  eye  to  one  edge  of  the  tree,  while 
the  indirect  line  of  sight  is  reflected  in  each  of  the  two  mirrors  to  the 
other  edge  of  the  tree.  That  the  two  lines  of  sight  are  parallel  and 
hence  that  the  distance  between  the  mirrors  is  equal  to  the  diameter 
of  the  tree  is  too  self-evident  to  demand  geometrical  demonstration. 

In  use  the  observer  holds  the  dendrometer  arm  horizontal  (if  the 
tree  is  in  the  normal  vertical  position)  with  one  of  the  mirrors  in  line 
between  his  eye  and  the  left-hand  edge  of  the  tree  at  the  point  to  be 
measured.  He  then  catches  the  reflection  of  the  second  mirror  in  the 
first,  thus  bringing  the  arm  into  a  line  perpendicular  to  the  line  from 
eye  to  tree.  By  sliding  the  second  mirror  in  or  out,  the  right-hand 
edge  of  the  tree  will  become  visible  in  it.  The  adjustment  is  now 
continued  until  the  left-hand  edge  as  seen  directly  and  just  above  the 
fixed  mirror,  and  the  right-hand  edge  as  seen  indirectly  through  the 
two  mirrors,  are  in  a  straight  line,  one  immediately  above  the  other. 
The  distance  between  the  mirrors  as  read  from  the  graduations  on  the 
arm  is  then  the  required  diameter. 

The  advantages  and  disadvantages  of  the  instrument  are  evident. 

a.  It  is  direct  reading. 

b.  The  distance  from  the  observer  to  the  point  observed  does  not 

have  to  be  determined. 

c.  As  a  result,  the  instrument  is  rapid  in  use. 

d.  It  may  be  set  for  a  given  diameter,  regardless  of  distance. 

e.  It  is  light  in  weight  and  of  convenient  shape  for  carrying; 

it  is  more  portable  than  a  pair  of  calipers  of  the  same  range. 


58  University  of  California  Publications  in  Agricultural  Sciences       [Vol.  3 

/.  It  will  measure  only  a  moderate  range  of  sizes. 
g.  While  very  accurate  for  a  hand  instrument,  it  is  not  capable 
of  extreme  precision. 
The  reason  for  the  last  two  statements  will  be  explained  in  the  fol- 
lowing pages. 

It  is  evident  that  it  will  meet  quite  well  the  requirements  already 
outlined.  It  fails  at  two  points  only — its  moderate  range  might  pre- 
vent its  use  in  very  large  timber,  and  its  lack  of  absolute  accuracy 
may  militate  against  it  for  very  precise,  scientific  work. 

Most  of  the  errors  of  such  a  dendrometer  are  easily  kept  negligible. 
Of  course  at  any  considerable  distance,  small  variations  of  diameter 
are  imperceptible  and  cannot  be  measured.  Since  the  minimum  visual 
angle  for  normal  eyes  is  one  minute,  two-tenths  of  an  inch  is  the 
smallest  variation  recognizable  at  a  distance  of  fifty  feet.  This  con- 
sideration applies  equally  to  all  dendrometers  which  do  not  involve 
telescopic  observations,  and  the  use  of  a  telescope  at  once  means  a 
heavy  and  awkward  instrument. 

If  the  arm  is  not  held  at  right  angles  to  the  direct  line  of  sight, 
the  graduations  on  the  arm  will  no  longer  measure  the  distance  between 
mirrors  along  the  indirect  sight  line,  nor  will  this  distance  agree  with 
the  desired  diameter.  However,  this  error  can  never  be  large  since, 
unless  the  arm  is  in  approximately  the  correct  position,  the  second 
mirror  cannot  be  seen  at  all  in  the  first,  and  to  center  its  image 
therein  is  an  instinctive  proceeding.  For  more  precise  work,  however, 
an  additional  aid  may  be  afforded  by  vertical  lines  scratched  into  the 
backing  of  each  mirror  at  its  exact  center,  which  are  to  be  brought 
into  apparent  coincidence  when  the  instrument  is  in  use.  An  alter- 
native method  of  obtaining  the  same  result  is  to  mask  the  fixed  mirror 
with  dark  paper  until,  at  the  most  convenient  distance  from  the  eye, 
the  whole  of  the  movable  mirror  can  just  be  seen  in  it.  The  position 
of  such  a  mask  is  shown  in  figure  2,  A. 

A  rotation  of  the  dendrometer  about  the  axis  of  the  arm  will,  of 
course,  raise  or  lower  the  indirect  sight  line  running  from  the  instru- 
ment to  the  tree.  Here  again,  however,  unless  the  position  is  essen- 
tially correct,  the  image  of  mirror  2  cannot  be  found  in  mirror  1.  The 
i-vmv  resulting,  moreover,  is  merely  the  amount  of  taper  that  occurs 
between  the  points  observed  by  the  direct  and  indirect  sight  lines, 
which  is  usually  negligible. 

Of  course,  if  the  two  lines  of  sight  are  not  parallel,  serious  errors 
will  result.     This  depends  on  having  the  two  mirrors  parallel  and  is 


1917] 


Bruce:  A  New  Dendrometer 


59 


in  part  a  matter  of  adjustment.  Two  opposed  adjusting  screws  must 
therefore  control  the  rotation  of  one  of  the  two  mirrors.  Adjustment 
is  simple.  Some  target  of  known  diameter  or  breadth  (a  sheet  of 
paper  against  a  dark  background  will  serve)  is  observed  with  the 
instrument  set  at  the  corresponding  diameter.  The  mirror  is  then 
rotated  by  its  adjustment  screws  until  the  two  edges  appear  in  line. 
This  process  is  delicate,  but  neither  complicated  nor  difficult. 


l""i"lT,,,|,,,i p"T"PT'»"T"i"T"i».rn 


|im^M|iiii|Mii,mijjmi|imjiiii|iii^i 


A~  Assembled  Dendrometer.  Elevation 

\rt 

» 

\j$y 

1       < 

w/  b 

B-  P/an 


C-  Cross  Section 
of  Arm 


D-  Slide   from  £e/ow 


1 I I 


Scale  for  A.B  an4C 

fnMl.M.i I 


0-rrn*| 


Scale  for  J).E.<*ndF 


_£■-  Slide.  Lie. vat/on 
a. — Arm. 

b. — Stationary  mirror  platform, 
c. — Sliding  mirror  platform. 
d.— Slide. 


Slide,  from  end 

e. — Fixed  mirror  and  support. 
f. — Adjustable  mirror  and  support. 
gg. — Adjustment  screws. 
h,  i. — Springs. 


Fig.  2 


The  one  error  which  dominates  all  others  is  that  due  to  a  failure 
of  the  arm  to  be  absolutely  straight.  This  is  unfortunately  a  matter 
of  instrumental  construction  and  not  of  adjustment,  and  the  difficulty 
of  making  this  arm  straight  is  surprisingly  great.  It  is  obvious  that 
almost  imperceptible  deviations  will  result  in  slightly  diverging  or 
converging  sight  lines  and  in  increasingly  serious  errors  in  the 
diameter  readings,  as  the  distance  at  which  the  measurement  is  taken 
is  lengthened.  In  the  instrument  described  the  maximum  error  from 
this  source  is  .6  inch  when  used  at  fifty  feet ;  it  is  doubtful  if  materially 
better  results  are  obtainable.  This  is  not  excessive.  Even  with  a  transit 
read  to  the  nearest  minute,  the  diameters  fifty  feet  away  can  be  read 


60 


University  of  California  Publications  in  Agricultural  Sciences       [Vol.  3 


but  to  the  nearest  .2  inch.  With  a  hand  instrument  of  the  common 
type  which  involves  the  measurement  of  the  angle  between  two  sight 
lines  it  is  difficult  to  provide  for  an  accurate  reading  closer  than  to 
the  nearest  10  minutes.  This  means  that  at  the  same  distance  1.7 
inches  would  be  the  minimum  recognizable  difference  in  diameter. 

Figure  2  shows  the  details  of  construction.1  a-a,  of  A,  B,  and  C, 
is  the  straight  arm  which  is  made  of  a  casting  of  aluminum  alloy. 
The  straight  edge  is  the  back  surface  of  the  slot  which  is  recessed  into 
the  upper  surface  of  the  arm,  as  is  best  seen  in  the  cross-section.  This 
cross-section  is  perhaps  unnecessarily  heavy,  but  was  so  designed  to 
insure  as  perfect  a  straight  edge  as  possible.  In  this  slot  travels  the 
slide  d  shown  in  detail  in  D,  E  and  F,  which  are  drawn  to  twice  the 
scale  of  A,  B  and  C.  This  slide  is  equipped  with  two  springs,  h  and  i, 
which  hold  it  against  the  back  and  upper  surfaces  of  the  slot.     Upon 


Fig.  3 

it  is  mounted  the  mirror  platform,  and  mirror  e  turned  to  an  angle 
of  45  degrees  to  the  axis  of  the  slide.  At  the  end  of  the  arm  a  second 
fixed  mirror  platform,  b,  is  mounted  on  which  is  the  second  mirror, 
f,  which  can  be  adjusted  by  the  two  opposed  adjusting  screws,  g-g. 
This  mirror  is  shown  with  both  center  line  and  mask,  though  both  are 
hardly  necessary.  The  scale  is  readily  seen  in  A.  This  is  read  by 
means  of  the  small  arrow  engraved  on  the  side  of  c,  as  shown  in  both 
A  and  E.    In  A  the  reading,  for  example,  is  12. 2- 

If  the  weight  of  the  instrument,  slightly  less  than  27  ounces,  is 
found  objectionable,  it  would  probably  be  safe  to  lighten  materially 
the  cross-section  of  the  arm  by  reducing  both  the  depth  of  the  down- 
\\;n(l  projecting  ribs  and  the  thickness  of  the  lateral  walls.  The 
mirrors  also,  as  shown,  are  very  generous  in  size,  and  might  be 
reduced  to  about  two-thirds  the  indicated  dimensions  without  intro- 
ducing any  serious  difficulties  through  restricting  the  field  of  vision. 


i  To  Mr.  V.  Arntzen  of  the  Civil  Engineering  Laboratory  of  the  University 
of  California,  credit  is  due  for  the  major  part  of  the  detail  of  design. 


1917]  Bruce:  A   New  Dendrometer  61 

The  instrument  shown  has  an  arm  eighteen  inches  long  and  will 
read  diameters  from  three  to  seventeen  inches.  A  longer  arm  is,  of 
course,  possible,  but  at  about  thirty  inches  a  point  is  reached  at  which 
the  adjustment  of  the  sliding  mirror  when  held  in  working  position 
would  become  awkward.  This  may  then  be  taken  as  the  practical 
limit,  unless  some  modification  of  the  principle  be  adopted.  This 
range  will  be  sufficient  for  a  great  deal  of  the  work  to  be  done.  If 
less  accuracy  is  required,  measurements  of  double  this  size  can  be 
secured  by  taking  the  center  of  the  tree  as  the  target  for  the  direct 
line  of  sight,  instead  of  the  left-hand  edge,  and  bringing  the  reflection 
of  the  right-hand  edge  in  line  with  the  center  point.  This  operation 
can  be  performed  more  accurately  than  might  at  first  be  supposed, 
and  the  method,  while  rough,  is  probably  quite  adequate  for  work  in 
connection  with  the  Pacific  coast  volume  tables  already  mentioned, 
in  which  taper  is  a  factor. 

A  quick  field  test  of  the  parallelism  of  the  sight  lines  consists  in 
measuring  the  same  diameter  at  two  different  distances.  The  readings 
should,  of  course,  be  identical,  or  rather,  since  a  small  observational 
error  is  unavoidable,  as  nearly  identical  as  would  be  two  consecutive 
measurements  from  a  single  position.  If  an  error  is  found  and  it  is 
not  convenient  to  make  the  proper  adjustment  it  may  be  simply  and 
quite  accurately  allowed  for,  by  taking  consecutive  observations  at 
two  known  distances.  For  example,  suppose  the  first  reading  is  14.8 
inches  and  the  second  reading  taken  at  one-half  the  distance  is  found 
to  be  14.4  inches.  Since  the  error  is  proportional  to  the  distance,  a 
reduction  of  the  distance  to  one-half  must  also  reduce  the  error  to 
one-half.  The  reduction  in  error  is  .4  of  an  inch,  the  total  original 
error  must  have  been  .8  of  an  inch,  and  the  correct  reading  is  therefore 
14.8  —  .8  =  14.0  inches.  Where  the  errors  are  small,  the  major  por- 
tion of  them  can  thus  be  eliminated,  even  if  the  distances  are  estimated 
instead  of  measured. 

A  modification  of  this  type  of  dendrometer  is  suggested  for  timber 
survey  crews  which  are  using  volume  tables  to  a  fixed  top-cutting  limit 
such  as  six  or  eight  inches.  All  that  is  necessary  in  such  cases  is  the 
pair  of  parallel  mirrors,  one  of  which  is  adjustable,  mounted  six  or 
eight  inches  apart  on  any  light  but  rigid  base  not  affected  too  readily 
by  changes  of  temperature  or  humidity.  By  thus  eliminating  the 
straight  edge  arid  slide  of  the  instrument  herein  described,  the  most 
serious  source  of  error  will  be  eliminated  and  the  cost  largely  reduced. 


